Average value Definition and 105 Threads

I have typed up the main problem in latex (see photo below) It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.

$\tiny{s8.5.5.2}$ Find the average value of the function on the given interval. $\sqrt{x}\quad [0,4]$ average value $\boxed{f_{ave}=\dfrac{1}{b-a}\int_a^b f(x) \ dx}$ so with $a=0$ and $b=4$ thus $\dfrac{1}{4-0}\displaystyle\int_0^4 \sqrt{x} \ dx \implies...

I tried using the equations above, but I wasn't really able to come up with an intuitive explanation. From my understanding, the electric field vector only varies in the x-y plane while the magnetic field vector only varies in the z-y plane. Also, both vary sinusoidally and both reach extrema...

While reading in the book of Introduction to Quantum Mechanics by David Griffith in the section of Fine structure of Hydrogen: spin- orbit coupling, he said that the average value of S operator is considered to be the projection of S onto J. I could not understand why he assumed that. please...

Another member and I, in private conversations, have been discussing the time average tension in a pendulum string. He has done a numerical analysis of the problem, and his calculations indicate that the time average tension is less than mg. I have analyzed the problem analytically by...

For spherical coordinate, ##dV = r^{2} \sin {\theta} dr d\theta d\phi## Therefore, $$<\frac{1}{r}> = \frac{1}{(\pi)(a_0)^{3}} \int_{0}^{\infty} {r e^{\frac{-2r}{a_0}} dr \int_0^{\pi} \sin {\theta}} d\theta \int_0^{2\pi} d\phi$$ From partial integral, I've found: $$\int_{0}^{\infty} r...

After getting the values of ψ₀(x) and ψ₁(x), I put them in the expression of ϕ(x) to get: ϕ(x) = (mw/πℏ)^(1/4) * exp[-(mw/2ℏ)x^2] * [α + βx√(2mw/ℏ)] Now when attempting to find the value of <x> by ∫xϕ(x) dx, I am having trouble determining the limits, as I am getting nothing useful by...

Does it make sense to just talk about the average value of a function without specifying the range over which the average is taken? It seems a common occurrence in discussions of waves to just mention that the average value of the complex exponential ##e^{ix}## is zero. But it will be zero only...

How do I calculate the average value of some damage values where there is say a 30% chance of critical hit and on a crit 150% damage is done? I run into this problem often, if I had to guess I'd say it's likely related to the binomial distribution because there's either a crit or not, fixed...

From the book Introduction to Quantum Mechanics by Griffiths,. In the section 6.4.1 (weak field zeeman effect) Griffiths tells that the time average value of S operator is just the projection of S onto J while finding the expectation value of J+S $$S_{avg}=\frac{(S.J)J}{J^2}$$ How to prove this?

https://imgur.com/a/Maq1N 1. Homework Statement Attached above is the problem statement. Essentially, they've given two different volumes involving some unknown function f(x) and ask that you find its average value over the given closed interval. Homework Equations Relevant equations include...

I was reading the statistical physics textbook and was really confused with the notation: I don't understand the last part of the section. Why is that \sum_{\sigma = \pm1} \sigma P(\sigma) equals to \left< \sigma \right>? And what does \left< \sigma \right> actually mean? Is it the average...

Homework Statement [/B] 1. Using the Schematic of a 555 in astable mode from the Wikipedia entry on '555 Timer IC," resistor R1 = 8 kOhms and capacitor C = 3 micro-Farads. Determine the value of R2 so that the output duty cycle is D = 0.7. Note that D is not expressed as a percentage here for...

Homework Statement Prove that A(X(ωk)2)≥A(X(ωk))2 if and only if X(ωk) has the same value for every k such that pk>0 for every category which actually occurs in the population Homework Equations A(X)=1/N∑nkX(ωk)=∑pkX(ωk) The Attempt at a Solution A[(X-A(X)2)]=A(X2)-A(X)2 and i believe the...

Hello. I was having a hard time determining how the formula for the average value of the waveform can be achieved. I've read that the ave since the waveform is symmetric is (Imin+Imax)/2. I want to know how to derive it using calculus. Thanks.

In kinetic theory, the number of molecules hitting a unit area of a surface per unit time with speeds between v and v + dv and angles between \theta and \theta + d \theta is found to be a function of sin(theta) and cos(theta). There will often be a practice problem asking to show that the...

Homework Statement The average value of the function ##g(x) = 3^{\cos x}## on the closed interval ##[- \pi, 0]## is: Homework Equations3. The attempt at the solution I used the standard method for finding average value over an interval with my calculator using an integral, and got the answer...

Hey everyone, I am preparing for a Calc 2 lecture tomorrow with KhanAcademy videos. I was watching a video on average value of a function in which he said, basically, you'll have some definite integral on a to b divided by (b-a). My question is, is this essentially calculating average speed? I...

Hello, I was wondering if someone out there could help me. The question: The speed, v metres per second, of a particle is given as v=8sin2t+3cos2t where t is in seconds. i) Find the total distance traveled in the time interval 0≤t≤1 ii) Find the average speed over this time interval.

I found two definitions of wigner function on space time. the first uses a Fourier transform of ##\rho (q+ y/2,q-y/2)## the second uses the Weyl transformation and parity operator ## exp (i \pi \theta N)## where N is the occupation number operator. Could you give me a link which shows the...

Homework Statement We know: In a measurement of quantum mechanics (basic) the average value is the cosine of the angle between the orthonormal basis of the measurement apparatus and the qubit (vector) entering it. Question: How do we prove this?Homework EquationsThe Attempt at a Solution Can...

Hello everyone. Let us consider two fermions whose spin state wave functions are certain (normalized) ##\psi_{1}## and ##\psi_{2}##. So the overall normalized spin state wave function of this system is of the form: ##\psi = (1/\sqrt{2})(\psi_{1} \psi_{2} - \psi_{2} \psi_{1})## Is there any...

Good morning everyone, I would really be pleased with a quick answer because I am very soon going to school. I have been asked to plot a graph, for that I am given a table with time in minutes from 0 to 6 on the first row. On the second row I was given activity in counts per minute with values...

Homework Statement In the given fig. we have to find average value of the voltage wave . So , for that we should take limits from 0 to pi/2.! But in solution it is taken as pi/2 to 3pi/2..! I did'nt get this..why so?[/B]Homework Equations average value of voltage = 1/pi limit integral...

So, while solving a problem a friend came up with involving the Totient function, I ended up doing a bit of research into the average asymptotics of the function. On page 268 of Introduction to the Theory of Numbers, it's mentioned that "The average of order of ##\phi\left(n\right)## is...

Homework Statement Consider a particle oscillating according to x(t) = a\cos(\omega t): Find \rho(x), the probability density to find particle at position x. Compute \langle x\rangle, \langle x^2\rangle. Homework Equations So in general we know that \textrm{Prob} = \int f_X...

Hey guys, I would greatly appreciate some help for this question I'm working on at the moment: "1.a) Use the values 9.1, 9.01, 9.001, 9.0001 to guess at the value of lim x->9 (x^3/2 - 27)/(x-9) 1.b) Then, use appropriate factoring to find the limit of (a)" For part a, I took sample values...

Homework Statement f(x,y) = ##e^{x+y}## D is the triangle vertices (0,0), (0,1) , (1,0) Homework Equations ##f(x,y)_{avg}=\frac{\iint_D f(x,y) dA}{\iint_D dA}## The Attempt at a Solution ##\iint_D dA \Rightarrow \int_{0}^{1}\int_{0}^{-y+1} dxdy = \frac{1}{2}## ##\iint_D...

Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: So for the first one, I just used f(b)-f(a)/ b-a and got 589, 1208, and 1366 respectively via simple substitution. For 1b, I...

This isn't exactly homework, for which I am not posting in the homework section. I am currently doing my undergrads, and am in the second semester of my first year. I am going to give my finals which start from tomorrow, with the first subject being Electrical Circuits II. I was solving the past...

Why we define that average value of some observable ##\hat{A}## in state ##\psi## is ##(\psi,\hat{A}\psi)## Why this isnot perhaps ##26(\psi,\hat{A}\psi)##?

The miles per gallon of my car on a recent trip at time t was given by $mpg(t) = 30 + 10sin(t)$ where $0 <= t <= \pi$. Find the average value of $mpg(t)$ over this time interval. So I'm kind of confused on how to do this. Here is what I have set up, let me know if I'm on the wrong track or not...

Homework Statement I have a question can the average value for an integral be negative. I don't see why not just checking. You know this evalutation f_ave = (1/b-a) ∫ f(x) dx Homework Equations thx The Attempt at a Solution

Homework Statement Suppose we walk along a unit semicircle. Homework Equations Find the average value of our distance from the base of the semicircle. The Attempt at a Solution y_{ave} = \frac{1}{1-(-1)}\int_{-1}^1\sqrt{1-x^2}\ dx = \frac{\pi}{4} OR y = \sin\theta, \qquad 0...

Homework Statement What is the average value of a function 1/x between x=2/3 and x=8/3? Homework Equations 1/(b-a) ∫f(x) dx with a and b being the lower and upper limits, respectively The Attempt at a Solution 1/([8/3] - [2/3])∫1/x dx 1/(6/3) ∫1/x dx 1/2 ∫1/x dx 1/2 * (ln x)...

Homework Statement Find expectation values ##\langle \hat{S}_x \rangle##, ##\langle \hat{S}_y \rangle##, ##\langle \hat{S}_z \rangle## in state ##|\psi \rangle =\frac{1}{\sqrt{2}}(|+\rangle +|- \rangle)## ##|+\rangle## and ##|-\rangle## are normalized eigen vectors of ##z## projection of...

Hello, for the Poisson problem Δu = -1 on a 2D circular disk with u = 0 on the boundary, we have average(u) = \frac{1}{8\pi}Area(disk), which is easy to see, as the solution is quadratic in the polar coordinate r. Does this (or a similar) relation hold for non-circular 2D domains? This...

Homework Statement Let f_k\rightarrow f in L^2(\Omega) where |\Omega| is finite. If \int_{\Omega}{f_k(x)}dx=0 for all k=1,2,3,\ldots, then \int_{\Omega}{f(x)}dx=0. Homework Equations The Attempt at a Solution I started by playing around with Holder's inequality and constructing...

Here is the question: Here is a link to the question: Find the average value of the function f on the interval [0,π]. (Enter your answer in an exact form.)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Calculate the average height above the x-axis of a point in the region 0 <= x <= 1 , 0 <= y <= x^2. I don't know what function to use to find the average height.

Homework Statement Since there is a graph and many things, I will post the link of my question. http://ibhl1-bccalculus.wikispaces.com/file/view/Solutions+to+optional+HL1+practice+for+final+(Larson+text).pdf Please scroll down to the last page (page 16) and my question is 6c. Homework...

Homework Statement Determine the average value of the following function over the interval [0,6] f(t) = (t2-1)e-0.5t Homework Equations 1/(b-a) ∫ f(x) = {f(x)} The Attempt at a Solution Substitution? let u = (t2-1) du/dt = 2t du/2t = dt 1/(6) ∫ u e-0.5t (du/2t)And here, I reach a...

Homework Statement Let f be a 2π-periodic function (can be any periodic really, not only 2π), and let g be a smooth function. Then lim_{n\rightarrow∞}\int^{B}_{A} f(nx)g(x) converges to \frac{1}{2π}\int^{2π}_{0}f(x) The Attempt at a Solution So far, I've come up with somewhat of...

I have the function: f(x,y)= x*(y^2)*e^-((x^2+y^2)/4), with x and y from -3 to 3 I took the integral of this function and got 0 as my answer. I need to find the average value, which is 1/area multiplied by the double integral. Since the double integral is 0, would the average value also be...

I put this under statistics because of your knowledge of the Gaussian. I have run into an elementary problem. I was considering what the average value, <x>, is for a Gaussian with an x offset, and got results which don't make sense to me. First, it is obvious that for P1(x)=e^(-(x^2)) the...

Homework Statement A particle of mass m moves in the potential energy V(x)= \frac{1}{2} mω2x2 . The ground-state wave function is \psi0(x)=(\frac{a}{π})1/4e-ax2/2 and the first excited-state wave function is \psi1(x)=(\frac{4a^3}{π})1/4e-ax2/2 where a = mω/\hbar What is the average value of...

Homework Statement find the average value of (1+4z)**3 on the surface ... Homework Equations The Attempt at a Solution All I really want to know is how to go about this? Is the average value on the surface just the surface integral over the area of (1+4z)**3?

As we know , the average value of a function over is : i wanted to calculate the average value of a function over its entire domain ( e.g ) firstly, i divided the interval into 2 intervals & and calculated the F_avg for each ,,, is my way correct ? thanks !

Homework Statement Find average value of Normal Acceleration from the information given. (See the picture) Homework Equations a^{→}_{ave}= Δv^{→}/Δt a_{t}=Δv/Δt The Attempt at a Solution I took the difference of the given angles in radians and divided it twice by Δt and get...

Hey so I am in a second semester calculus class and we are learning about the average value of the function. We are presented the following formula: (1/(b-a)) ∫a to b f(x) dx for a ≤ x ≤ b This is basically doing (x1+...xn)/n, except for infinitely many numbers over the interval...